Prediction of thermal protection systemmaterial permeability and hydraulic

tortuosity factor using Direct SimulationMonte Carlo

Revathi Jambunathan∗, Deborah A. Levin†, Arnaud Borner‡,

Joseph C. Ferguson§, and Francesco Panerai¶

Carbon preforms used in Thermal Protection System (TPS) materialsare 80 to 90% porous, allowing for boundary layer and pyrolysis gases toflow through the porous regions. The bulk material properties such aspermeability and hydraulic tortuosity factor affect the transport of theboundary layer gases. The use of Direct Simulation Monte Carlo alongwith the Klinkenberg permeability formulation allows us to compute thecontinuum permeability and Knudsen correction factor for flow in the tran-sition regime. In this work, we have computed the permeability for twotypes of carbon preforms, namely, Morgan Felt and FiberForm, and as-sessed the effect of orientation on the permeability. Since both the mate-rials are anisotropic, the permeability was found to depend on orientation,wherein, the materials are more permeable in the in-plane orientation thanthe through-thickness orientation. The through-thickness orientation wasalso more tortuous compared to the in-plane material orientation. Com-pared to Morgan Felt, FiberForm is less permeable, in both, through thick-ness and in-plane directions.

∗Graduate Student, Department of Aerospace Engineering, University of Illinois, Urbana-Champaign,IL-61801, AIAA Student Member.,†Professor, Department of Aerospace Engineering, University of Illinois, Urbana-Champaign, IL-61801,

AIAA Fellow‡Reseach Scientist, STC at NASA Ames Research Center, MS258-6, Moffett Field, CA, 94035, USA,

AIAA Member§Junior Research Scientist, STC at NASA Ames Research Center, MS258-6, Moffett Field, CA, 94035,

USA¶Research Scientist, AMA Inc. at NASA Ames Research Center, MS234-1, Moffett Field, CA, 94035,

USA, AIAA Member.

1

https://ntrs.nasa.gov/search.jsp?R=20190026637 2020-03-17T22:08:10+00:00Z

I. Introduction

Spacecraft are exposed to aerothermodynamic heating during entry into a planetary at-

mosphere and require thermal protection systems (TPS) to shield the vehicle and its crew

from the chemically reacting high temperature gases. Candidate TPS materials, such as

the Phenolic Impregnated Carbon Ablator (PICA)1,2 are composed of a complex network

of carbon fibers of micrometer size impregnated with a phenolic resin. These TPS mate-

rials undergo ablation, a self-sacrificial process, to protect the vehicle. Characterizing the

material morphology and predicting its permeability and tortuosity factor is critical in mod-

eling the transport of high temperature gases through the material, as well as the material

response. The main objective of this work is to compute the material permeability and hy-

draulic pressure driven tortuosity factor of porous preforms used in TPS materials, namely

Morgan carbon felt3 (Morgan Advanced Materials, Fostoria, Ohio,USA) and FiberFormr3

(Fiber Materials, Inc.). In addition we aim at determining the representative elementary vol-

ume (REV) for each microstructure so that, material analysis can be performed on smaller

samples without compromising on the accuracy of the computed material properties. We

will analyze the effect of material orientation on the material properties, since these fibrous

microstructures are known to be anisotropic.4,5 The small characteristic length scale of this

problem renders the flow to be at non-continuum regimes even at atmospheric pressures

requiring the use of more accurate kinetic theory based approaches to solve the transport

within porous microstructure.

Direct Simulation Monte Carlo6 (DSMC) is a probabilistic particle-based method that

provides a numerical solution to the Boltzmann transport equation which accurately models

finite Knudsen number flows. The gas flow is simulated by computational particles, where

each particle represents a large number of real atoms or molecules. Traditionally, the compu-

tational domain is divided using a uniform Cartesian grid such that the cell size is less than

2

the local mean free path, in order to satisfy the DSMC criteria. At every timestep, particles

are moved, sorted into nearest neighbors or cells, and binary collisions are performed between

particles that belong to the same cell. For problems with sharp density gradients, a uniform

grid would have to refine the cells everywhere in the domain, thereby over-refining the cells

in some regions and increasing the computational time. To reduce the computational time,

more recently, adaptive mesh refinement has been implemented, where cells are refined only

in the regions where the local mean free path is small. However, for accurately modeling flow

through highly irregular bodies, an octree-based solver is an advantageous approach when

combined with additional capabilities such as volume-of-fluid and ray-tracing to determine

the exact fluid volume of cut-cells and for gas-surface interactions. Even with efficient oc-

tree approaches, performing DSMC calculations for flow through fibrous microstructures is

computationally expensive and novel parallelization strategies are required to reduce sim-

ulation run-time. In an effort to improve the computational efficiency, we have developed

a scalable three-dimensional DSMC solver called Cuda-based Hybrid Approach for Octree

Simulations (CHAOS),7 to accurately model the flow through irregular porous media and

exploit the heterogeneous architectures available in many petascale supercomputers.8 In the

CHAOS DSMC solver, linearized octrees are employed to efficiently account for multiple

length scales. Large number of gas-surface interactions between the gas and TPS mate-

rial are modeled using ray-tracing algorithms, using massively parallel methods on multiple

GPUs. In our previous work,7 we have shown 85% strong scaling and nearly 100% weak

scaling with CHAOS for flow through the fibrous microstructures.

II. Numerical Method and Test Cases

The target problem involves computing morphological properties of two materials, namely,

FiberForm and Morgan felt.3,5, 9 Figure 1 shows a computational model of the felt and

3

FiberForm TPS materials. They were obtained using a tomography image that was surface

rendered and converted into a standard stl format with an average of 4 million triangular

surface elements. These models allow us to compute permeability,9–11 which is an impor-

tant material property that controls the transport process of boundary layer and pyrolysis

gases, which in turn enables the prediction of heat transfer to the material and the resulting

degradation. Similarly, the hydraulic tortuosity factor12–14 is a critical geometrical param-

eter as it quantifies how the gas particles deviate from a straight path to flow through the

irregular pores of the material in a pressure driven flow. This hydraulic pressure driven tor-

tuosity factor can be used to determine the hydraulic diameter15 of porous materials, which

in turn is used to obtain the Reynolds number and friction factor15 in pressure driven flow

through porous media. From micro-tomography measurements, Panerai et al.,9 showed that

the porosity for FiberForm is nearly 85-91% and that of the felt is 94%. Borner et al.,5 used

DSMC simulations to compute the permeability of the fibrous microstructures, predicting

a higher permeability for the Morgan felt as compared to FirberForm. It was also shown,

both from experiments9 and previous DSMC5 computations that both, the Morgan Felt

and FiberForm material have a transverse isotropic structure, that is, the through-thickness

(TT) orientation is less permeable than the in-plane (IP) direction. That is due to the

manufracturing process of these materials, in which the billet is compressed in one direction.

The DSMC simulations were performed on a smaller sample size of (520×520×520) µm3.

However, in the current work, due to the capability of CHAOS DSMC solver to handle large

sample sizes, we will compute the material permeability and hydraulic tortuosity factor on

a (1×1×1) mm3 material sample and also determine the REV, that is, the smallest sample

size that can be used for analysis without significantly affecting the predicted macroscopic

transport properties of the material.

The definition of the DSMC test cases performed on the two materials, Morgan felt and

FiberForm, to study the effect of temperature, sample size and orientation, are given in

4

(a) Morgan felt microstructure reconstructed using 2.6million surface mesh elements.

(b) FiberFormr microstructure reconstructed using 5.6million surface mesh elements.

Figure 1: Fibrous microstructure reconstructed from tomography images of candidate TPSmaterials. Red to blue color scale represents increasing distance from the top surface. TheTT direction is along the z-axis, and the IP direction is along the x or y-axis.

Tabs. 1 and 2. Test cases Felt 1 and 2 were performed on the large 1 mm3 material to

study the effect of orientation on continuum permeability. To study the effect of sample

size, the Ko,TT obtained for the large sample size in test case Felt 1 is compared with the

DSMC calculations performed by Borner et al.,5 for a smaller sample size of (520×520×520)

µm3. The DSMC computations performed by Borner et al.,5 used an initial gas and surface

of 300 K, while, in contrast, the CHAOS DSMC simulations performed in this work use

different initial gas and surface temperature at 2000 and 300 K, respectively. However, since

in the simulations performed in this work, the material surface was maintained at 300 K

throughout the DSMC computation and the gas-surface interactions were fully diffuse, the

gas temperature within the porous material is expected to equilibrate with that of the surface

temperature. For the FiberForm material, four test cases are performed using CHAOS

DSMC as shown in Table 2. Test cases Form 1 and 2 are performed for the large 1 mm3

sample with different initial gas and surface temperature to study the effect of orientation

on the continuum permeability. To study the effect of initial gas and surface temperature on

the TT continuum permeability, test case Form 3 is performed with initial gas and surface

5

temperature at 1319 K, similar to the permeability experiments performed by Panerai et

al.,9 on a 22 mm3 sample. Additionally, test case Form 4 is performed for a sample size of

520 µm3 with initial gas and surface at 1319 K, to study the effect of sample size on the TT

continuum permeability, compared to Form 3.

Table 1: Test case definition for Morgan felt

Test Case Orientation Sample Size (µm3) Initial Tg and Ts

Felt 1 TT 1032×1032×1032 Tg=2000 K, Ts=300 K

Felt 2 IP 1032×1032×1032 Tg=2000 K, Ts=300 K

Table 2: Test case definition for FiberForm

Test Case Orientation Sample Size (µm3) Initial Tg and Ts

Form 1 TT 1032×1032×1032 Tg=2000 K, Ts=300 K

Form 2 IP 1032×1032×1032 Tg=2000 K, Ts=300 K

Form 3 TT 1032×1032×1032 Tg=1319 K, Ts=1319 K

Form 4 TT 520×520×520 Tg=1319 K, Ts=1319 K

For all the above defined test cases, argon gas particles are introduced at the domain inlet

with an initial bulk velocity of 460 m/s in the negative z-direction for the TT calculations and

in the +x-direction for the IP computations, with an initial gas temperature, Tg. Since the

geometry is irregular, non-periodic, and represents only a small sample of the bulk material,

a pseudo-periodic boundary condition, also called mirror boundary condition, is imposed on

the cross-stream boundaries. For this boundary condition, when particles cross the boundary

after movement, they are specularly reflected back into the domain, instead of allowing them

to enter from its opposite plane, which we call as the periodic counterpart. This pseudo-

periodic assumption holds true in our simulations because, on average, the total number and

total energy of particles that leave the domain from one cross-stream boundary is equal to

the total number and total energy of outgoing particles from its periodic counterpart. For

all the DSMC simulations, the material microstructure is assumed to be stationary with a

6

constant surface temperature, Ts. When gas particles collide with the material surface, they

are reflected diffusely with energy equivalent to the surface energy.

III. Material Permeability and Hydraulic Tortuosity Results

To enable the comparison of fully kinetic simulation results with the experiments9 of gas

transport through porous media, the Klinkenberg formulation10,11 is used to estimate the

effective permeability, K, of the TPS material, which accounts for rarefied slip effects, and

is given as11,

K = Ko[1 + (b/Pav)], (1)

where, b is the permeability slip parameter or Knudsen correction factor that varies with

temperature and gas composition, Ko is the continuum permeability, and Pav is the average

gas pressure in the domain. Ko is strictly a function of the material microstructure, while

b depends on both, the microstructure and the gas flow conditions. From Darcy’s law,

continuity equation, and the ideal gas law, the mass flow rate and permeability are related

as follows,11

F =µmRTL

AM∆P= Ko(Pav + b) (2)

where, the permeability force F can be used to graphically relate Ko, Pav, and b by substi-

tuting the values of mass flow rate, m, temperature, T , and the pressure difference between

the inlet and outlet, ∆P , obtained from the DSMC calculations. Since the viscosity coef-

ficient, µ, molar gas constant, R, the molar mass, M , of the gas species, the length L and

cross-sectional area of the domain, A, are known, the values of Ko and b are obtained from

a linear least-squares fit of F vs Pav values from DSMC calculations performed for different

7

average pressures. The continuum permeability, Ko, is computed by taking the slope of F vs

Pav and the value y-intercept gives the permeability slip parameter. The values of Ko and b

obtained for the test cases Felt 1 and 2, and for cases Form 1-4, are discussed in subsections

III A and B.

A. Effect of orientation, size and temperature on Permeability of Morgan Felt

DSMC simulations are performed with different average pressures for the 1 mm3 Morgan felt

material, in both the through-thickness (TT) and in-plane (IP) directions. The variation

of the permeability force, F , obtained by substituting values from the DSMC simulations

in Eq. (2), for different average pressures are shown in Figs. 2(a) and 2(b) for test cases

Felt 1 and Felt 2, respectively. The continuum permeability of the material is obtained

from the slope of the line and the y-intercept gives the Knudsen correction factor, b. It is

found that the continuum permeability, Ko, of Morgan felt, is 209.9×10−12 m2 in the TT

direction, with b =7027.15 Pa. In the IP direction, the continuum permeability is higher,

Ko = 271.8 × 10−12 m2 with b =5896.69 Pa, as compared in Tab. 3. The permeability is

higher in the IP direction compared to the TT direction due to the anisotropic alignment of

the fibers. Since the permeability is higher in the TT direction, the number of gas-surface

interactions in TT are larger than in IP, thereby increasing the Knudsen correction factor,

b, in the TT direction.

The TT continuum permeability obtained for the large 1 mm3 felt material is compared

to the TT continuum permeability obtained by Borner et al.,5 for a smaller material volume

of (520×520×520) µm3 in Tab. 3. Note that, Borner et al.,5 performed DSMC simulations

with equal gas and surface temperature at 300 K, whereas for the Felt 2 case the initial

gas temperature used was 2,000 K with surface temperature equal to 300 K. It can be ob-

served that the continuum permeability of Morgan felt obtained for the larger material from

CHAOS DSMC is approximately 7% higher than that obtained from the DSMC computation

8

(a) F versus average pressure in the TT direction.Ko,TT =209.9×10−12 m2, b=7027.15 Pa.

(b) F versus average pressure in the IP direction.Ko,IP =271×10−12 m2, b=5896.67 Pa.

Figure 2: Continuum permeability Ko and Knudsen correction factor, b, of Morgan Felt inTT and IP.

performed by Borner et al.,5 for a smaller material sample. Since the difference in material

permeability is within the uncertainty of ±10%, a smaller material sample of 520 µm3 may

be sufficient to compute the material permeability for Morgan felt. The Knudsen correction

factor, however, is higher in CHAOS compared to the results obtained from Borner et al.5

Unlike the continuum permeability, which is a material property, the b factor is affected by

the gas temperature, which is approximately 300 K within the material pores, but higher at

the inlet of the computational domain and at the leading edge of the material. As a result,

the Knudsen correction factor, b, obtained from the Felt 1 simulations is higher than that

obtained from Borner et al.,5 even though the material temperature in both the calculations

is 300 K. According to the Klinkenberg theory, the b values from higher temperature simu-

lations must scale with the lower temperature b∗ calculations as bb∗

= µ√T

µ∗√T

∗ . Borner et al.5

have shown that for simulations with different material temperatures, the scaling ratio of

b/b∗

(µ√T )/(µ∗

√T ∗)

is close to unity, i.e., the ratio of b/b∗ is equal to the ratio of (µ√T )/(µ∗

√T ∗).

Applying this scaling relationship,5 we find that substituting the b value obtained from

9

CHAOS for the TT Morgan felt calculations and the average of the initial gas and surface

temperature, i.e., T=(2000+300)/2=1150 K, the ratio of b/b∗

(µ√T )/(µ∗

√T ∗)

= 0.93. Thus, the b

obtained from CHAOS satisfies the temperature and viscosity scaling ratio within the 10%

uncertainty error involved in the material characterization studies.

Table 3: Permeability comparison of Morgan felt in IP and TT

Solver CHAOS Felt 2(IP)

CHAOS Felt 1(TT)

Borner et al.5

(TT)

Sample size (µm3) 1032×1032×1032 1032×1032×1032 520×520×520

Surface Temperature (K) 300 300 310

Initial gas temperature (K) 2000 2000 310

Ko×10−12m2 271.8 209.9 195

Knudsen correction factor, b (Pa) 5,896.69 7,027.15 1,403

B. Effect of orientation and sample size on permeability of FiberForm

The variation of the permeability force, F , with average pressure for the 1 mm3 Form 1

case is shown in Fig. 3(a). The TT continuum permeability obtained by taking the slope

of the F vs Pav line for Form 1 case is 31.96×10−12m2 and the Knudsen correction factor,

b=23,357.3 Pa. Comparing our results with those from Panerai et al.,9 it is found that

the continuum permeability predicted from the CHAOS DSMC simulations with 1 mm3 is

38% smaller compared to the Ko from the experiments that used a 22 mm3 sample. This

difference in the Ko,TT values from DSMC and experiments may be due to the large mi-

crostructure variablity of FiberForm and due to the possible difference in the microstructure

of our computational sample and the large specimen used in the experiments. Experimental

data9 have also shown that the uncertainty in the material permeability are up to ± 10%

and that long-range variabilities in the fibrous structure within the material may affect the

TT material properties.

Figure 3(b) shows the variation of F with average pressure for the 1 mm3 FiberForm in

10

(a) F versus average pressure in the TT direction.Ko,TT =31.96×10−12 m2, b=23,357.3 Pa.

(b) F versus average pressure in the IP direction.Ko,IP =85.84×10−12 m2, b=10,350.69 Pa.

Figure 3: Continuum permeability, Ko and Knudsen correction factor, b, of cases Form 1and Form 2 in TT and IP.

the IP direction, i.e., for Form 2 test case. The slope of this line resulted in an IP Ko of

85.84×10−12 m2 and b=10,350.69 Pa. Due to the anisotropic fiber orientation, the material

permeability in the IP direction is nearly 2.5 times higher than the TT Ko. Similar to the

Morgan felt calculations, the less permeable TT direction results in more gas-surface inter-

actions, higher slip correction and therefore higher b values compared to the more permeable

IP orientation. The Ko and b values from Form 2 case are compared in Table 4 with the

values obtained from the experiments9 as well as the DSMC computations performed by

Table 4: Permeability comparison of FiberForm in the In-Plane (+x) direction

Solver CHAOS Form 2case

Experiments Borner et al.5

Sample size (mm) 1.032×1.032×1.032 21×21×21 0.52×0.52×0.52

Initial Tg and Ts Tg=2000 K,Ts=300 K

Tg=300 K,Ts=300 K

Tg=310 K,Ts=310 K

Ko×10−12m2 in TT 85.84 112 97.54

Knudsen correctionfactor, b (Pa) in TT

10,350.69 1,408 1,517.5

11

Borner et al.5 on a (520µm)3 material sample. It was found that the IP Ko from CHAOS

for the 1 mm3 sample is 24% smaller than the IP Ko obtained from the experiments,9 and

12% smaller than the results obtained from the DSMC computations performed by Borner

et al.5 The Knudsen correction factor, b, from CHAOS simulations is higher than the b

value obtained from the experiments as well as the DSMC computations performed with

Tg=Ts=300 K. This is because, b depends on the gas temperature and for the Form 2 test

case, the initial gas and surface temperatures were 2000 and 300 K, respectively, as shown

earlier in Tab. 1.

Table 5: Permeability comparison of FiberForm in the through-thickness (-z) direction

Solver CHAOSForm 1 case

CHAOSForm 3 case

CHAOSForm 4 case

Borner etal.5

Sample size (µm3) 1032×1032×1032 1032×1032×1032 520×520×520 520×520×520

Initial Tg (K) 2000 1319 1319 1319

Surface temp. (K) 300 1319 1319 1319

Ko×10−12m2 31.96 32.95 38.81 52.08

b (Pa) 23,357.3 45,402 40,118 12,198

To determine the effect of gas temperature on the continuum permeability, we performed

simulations for the Form 3 case, with initial gas and surface temperature at 1319 K, similar

to the DSMC calculations performed by Borner et al.5 in the TT direction. The F versus

average pressure values obtained for this thermal equilibrium Form 3 case are shown in

Fig. 4. When the initial gas and surface temperature are equal, the continuum permeability

in the TT direction was 32.95×10−12 m2, which is similar to the Ko obtained for Form 1 case

with different Tg and Ts shown in Fig. 3(a) and compared in Table 5. It is not surprising

to see that the bulk continuum permeability of the material remains unchanged with initial

gas or surface temperature, since it is strictly a microstructure dependent material property.

But, as discussed earlier, the Knudsen correction factor, b, depends on the gas properties

and we see that for the Form 3 case with Tg=Ts=1319 K, the b value is nearly twice that

12

of the Form 1 case with Ts at 300 K and initial Tg=2000 K. Using Form 3 reference values

of b∗ = 45, 402 Pa and T ∗ = 1319 K, and Form 1 values of b = 23, 357 K, and T =

(1/3) ∗ 2000 + (2/3) ∗ 300 = 866.6 K, in the scaling ratio, we find that b/b∗

(µ√T )/(µ∗

√T ∗)

=

0.89, which is 11% below unity, or close to the 10% uncertainty of material characterization

studies. Note that, the temperature T used in the computation of the scaling ratio is a

weighted average of the initial gas temperature of Tg = 2000 K and surface temperature of

Ts = 300 K. The weights for the gas and surface temperature are 1/3 and 2/3 respectively,

because the region of the computational domain consisting of the diffusely reflecting material

surface is 2/3rd of the computational domain, resulting in gas temperature equal to 300 K

in that region. Since Morgan felt is more permeable that Fiberform, is allows the high

temperature gases to penetrate through the material more easily. Therefore, a weighting

factor of 1/2 was used to compute the average temperature used in the scaling relation for

the Felt 1 calculations, discussed earlier in Sec. III A.

Figure 4: F versus average pressure for 1 mm3 FiberForm in the TT direction forTgas=Tsurf=1319 K, i.e., Form 3 case. Ko,TT=32.95×10−12 m2, b=45,402 Pa.

To understand the effect of sample size on the permeability calculations, we performed

13

DSMC simulations i.e., Form 4 case, with a smaller material sample of (520×520×520) µm3

with initial gas and surface temperature at 1319 K. The variation of F values with average

pressure for Form 4 case is compared with the experiments performed for 22 mm3 sample

with the same gas and surface temperature in Fig. 5. The continuum permeability obtained

for the Form 4 case is Ko=38.81×10−12 m2 which is nearly 25% less than the Ko obtained

from the experiments in the TT direction. This difference in TT permeability obtained from

the computations and experiments may be due to the large density variation of the material

in the TT direction.9 Further, the permeability force, F , obtained from the CHAOS DSMC

calculations is higher than that obtained from the experiments for a given average pressure.

For the same gas and surface temperature as well as material sample size of 520 µm3, Borner

et al.5 obtained a value of 52.08×10−12 m2 from their DSMC simulations, similar to the

experiments which used an even larger material sample of 22 mm3.

Figure 5: Variation of F with average pressure for 520 µm3 FiberForm sample in the TTdirection for Tgas=Tsurf=1319 K, i.e., Form 4 case. Ko,TT=38.81×10−12 m2, b=40,118 Pa.

14

C. Hydraulic tortuosity factor in a pressure driven flow

The hydraulic tortuosity, τh, is defined as the square of the ratio of the effective tortuous

gas streamline through the material, lt, to the shortest straight path, l, which is also equal

to the depth of the material, in cases discussed here. This ratio is equivalent to the velocity

ratio,12–15

(τh)i =

(ltl

)2

=

(< u >

< ui >

)2

(3)

where, < u > is the average speed of the gas particles, and < ui > is the average value of the

velocity component in the ith direction, within the material. From the steady-steady results

obtained from the CHAOS DSMC simulations, u and ui are sampled in each leaf node,

where u = (u2x + u2y + u2z)(1/2). By taking the average of < u > and < ui > over all the cells

within the material, the bulk hydraulic tortuosity factor for the ith direction is computed by

substituting the sampled values of < u > and < ui > in Eq. 3. The variation of τh with

average pressure within the 1 mm3 FiberForm and Morgan felt materials in the TT and IP

orientation are compared in Fig. 6. For both, Morgan felt and FiberForm, the TT orientation

is more tortuous compared to the IP orientation. It can be observed that the effective path

of the gas is more tortuous through the rigid FiberForm microstructure compared to the

Morgan felt microstructure. This is consistent with the smaller permeability obtained in the

TT direction compared to the IP direction. In addition, the hydraulic tortuosity factor is

found to increase with increasing average pressure. As the flow becomes more continuum-

like, the twistiness of the effective streamline path is larger due to smaller velocity slip at

the material surface, compared to the lower pressure cases. At lower pressure the higher slip

effects reduce the curvature of the effective streamline and the effective tortuosity. It was also

found from Fig. 6 that the hydraulic tortuosity factor, τh remained constant above 4,000 Pa.

When the pressure increases, the gas-gas collisions dominate compared to the gas-surface

15

Figure 6: Variation of hydraulic tortuosity factor, τh with average pressure within thematerial.

collisions, the effective flow path does not change, maintaining the τh at a constant value

beyond 4,000 Pa. The length of the streamlines, lt, was also computed using the commercial

software, Tecplot, and the ratio of (lt/l)2 resulted in tortuosity factors within 2% of that

obtained from the square of the velocity ratio. Since the hydraulic tortuosity factor, τh,

Table 6: Comparison of hydraulic and diffusive tortuosity factors

Tortuosity CHAOSτh=(< u >/< ui >)2

Panerai et al.4 τd

FiberForm (TT) 1.25 1.225

FiberForm (IP) 1.18 1.15

Morgan felt (TT) 1.12 1.08

Morgan felt (IP) 1.08 1.053

from the simulations did not vary beyond 4,000 Pa, the τh, for the highest pressure case is

compared with the continuum diffusion tortuosity factor, τd obtained by Panerai at al.4 in

Tab. 6. It can be seen that the maximum difference between the τh computed using the

velocity ratio method and the diffusive tortuosity factor, τd is within 3%.

16

Table 7: Tortuosity comparison for FiberForm in the through-thickness direction, usingvelocity ratio and particle tracking

Pressure (Pa) Velocity Ratio,τh=(< u >/< ui >)2

Particle Tracking, ld/l

1012 1.18 2.22

9018 1.25 3.4

In addition, 100,000 gas particles were tracked from the time they enter the FiberForm

material in the TT direction, to the time they leave the material, to determine the average

length of their kinetic trajectories throughout the material. The ratio of average length

of the particles trajectories, ld, to the length of the material, l, for FiberForm at 1,012

and 9,018 Pa average pressures are compared with the TT hydraulic tortuosity factor, τh,

in Tab. 7. It was found that, on average, the gas particle trajectory is 2.2 and 3.4 times

longer than the material length, for 1,012 and 9,018 Pa pressures, respectively. The length

of the particle trajectory is found to increase with increase in the pressure, similar to the

trend observed for the τh, but, the ratio of kinetic trajectory length to material size is larger

than τh. The particle trajectories includes both gas-gas and gas-surface collisions, and all

the displacements it undergoes when trapped within the material, whereas the hydraulic

tortuosity factor, τh, only accounts for the effective length of the streamline as the gas flows

through the porous region due to pressure gradient.

IV. Conclusion

The permeability of hydraulic tortuosity factor of two types of carbon preforms, namely,

Morgan Felt and FiberForm, are computed using a multi-GPU hybrid MPIC-CUDA Direct

Simulation Monte Carlo (DSMC) solver. It was found that for both the anisotropic ma-

terials, the in-plane orientation is more permeable than the through-thickness orientation.

Compared to Morgan Felt, FiberForm is less permeable, in both, through thickness and

17

in-plane directions. On the contrary, the Knudsen correction factor is higher for Fiberform

compared to the Felt, because, the gas undergoes higher gas-surface interactions in the less

permeable Fiberform, resulting in higher slip, and therefore higher correction factors. The

hydraulic tortuosity was higher for the FiberForm sample compared to Morgan Felt sample,

and it was also found that the through-thickness orientation is more tortuous compared to

the in-plane orientation.

Acknowledgments

We are grateful for the funding support provided by NASA through the Grant NNX

16AD12G. This research is part of the Blue Waters sustained-petascale computing project,

which is supported by the National Science Foundation (awards OCI-0725070 and ACI-

1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois

at Urbana-Champaign and its National Center for Supercomputing Applications.

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